This paper proposes a general and unified framework for testing the equality of a broad class of parameters, defined as a smooth function of expectations of symmetric kernels, across multiple independent populations. We consider two test statistics, a Wald-type statistic and an ANOVA-type statistic. The asymptotic distribution of the first one is derived under a fixed-dimension regime, whereas the second one is studied under both fixed and increasing-dimension regimes, where the parameter dimension diverges with the sample size. Based on these limiting distributions, we construct test procedures enabling asymptotically exact inference without parametric assumptions. Additionally, an alternative null distribution estimator based on a weighted bootstrap approximation is studied, which is applicable to the ANOVA-type statistic under a fixed-dimension regime. The finite-sample performance and computational efficiency of the proposed procedures are evaluated through an extensive simulation study and a real dataset application.

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